Simple Linear Regression Analysis
2. Weight of Car: Miles gallon – Do heavier cars really use more gasoline? The following data were obtained from Consumer Reports (Vol 62 no. 4). Weight of car is in hundreds of pounds.
Car Weight 27 44 32 47 23 40 34 52
MPG 30 19 24 13 29 17 21 14
A simple linear regression of the model MPG = b + b WEIGHT
The results are shown below:
MPG & CAR WEIGHT
REGRESSION FUNCTION & ANOVA FOR MPG
MPG = 43.32625 - 0.600702 WEIGHT
R-Squared = 0.895426
Adjusted R-Squared = 0.877997
Standard error of estimate = 2.236055
Number of cases used = 8
Analysis of Variance
p-value
Source SS df MS F Value Sig Prob
Regression 256.87 1 256.87 51.37567 0.000372
Residual 29.99 6 4.99
Total 286.875 7
MPG & CAR WEIGHT
REGRESSION COEFFICIENTS FOR MPG
Two-Sided p-value
Variable Coefficient Std Error t Value Sig Prob
Constant 43.32625 3.23051 13.41156 0.000011
WEIGHT -0.60070 0.08381 -7.16768 0.000372 *
Standard error of estimate = 2.236055
Durbin-Watson statistic = 0.995097
(a) What is the estimated simple linear regression?
(b) What sort of relationship exists between MPG and car weight? Does the relationship make sense to you? Why or why not?
(c) Interpret the R-square in the computer output.
(d) Estimate the MPG for a car that weighs 20 hundred pounds and a car that weight 60 hundred pounds.
(a) The estimated simple linear regression equation is given as:
MPG = 43.32625 - 0.600702 * WEIGHT
(b) The relationship between MPG and car weight can be interpreted as follows: for every 100-pound increase in car weight, the MPG value is estimated to decrease by approximately 0.6007. This means that heavier cars tend to use more gasoline. In this case, the negative coefficient indicates an inverse relationship between car weight and MPG.
This relationship does make sense as heavier cars generally require more energy to move, resulting in lower fuel efficiency and, therefore, lower MPG.
(c) The R-squared value in the computer output is 0.895426. This value represents the goodness-of-fit of the regression model. It indicates that 89.5% of the variation in MPG can be explained by the car weight variable in this dataset. In other words, the car weight variable has a strong correlation with MPG, explaining a significant portion of the variability observed in MPG values.
(d) To estimate the MPG for a car that weighs 20 hundred pounds, we can substitute the weight value (20) into the regression equation:
MPG = 43.32625 - 0.600702 * 20
Calculating this equation, we find:
MPG ≈ 43.32625 - 12.01404
MPG ≈ 31.31221
Therefore, the estimated MPG for a car weighing 20 hundred pounds is approximately 31.31.
For a car that weighs 60 hundred pounds, we can similarly substitute the weight value (60) into the regression equation:
MPG = 43.32625 - 0.600702 * 60
Calculating this equation, we find:
MPG ≈ 43.32625 - 36.04212
MPG ≈ 7.28413
Therefore, the estimated MPG for a car weighing 60 hundred pounds is approximately 7.28.
(a) The estimated simple linear regression is given by the equation:
MPG = 43.32625 - 0.600702 * WEIGHT
This equation represents the relationship between MPG (Miles per Gallon) and the weight of the car. The coefficient of the WEIGHT variable (-0.600702) indicates the change in MPG for every one unit increase in the weight of the car.
(b) The negative coefficient (-0.600702) suggests that as the weight of the car increases, the MPG decreases. This means that heavier cars tend to use more gasoline per mile. The relationship between MPG and car weight does make sense; intuitively, it is expected that heavier cars require more fuel to move due to increased energy required to overcome the increased mass.
(c) The R-squared value in the computer output is 0.895426. R-squared provides an indication of the proportion of the variation in the dependent variable (MPG) that can be explained by the independent variable (car weight) in the regression model. In this case, an R-squared value of 0.895426 means that approximately 89.54% of the variation in MPG can be explained by changes in car weight. This indicates a strong relationship between MPG and car weight.
(d) To estimate the MPG for a car that weighs 20 hundred pounds, substitute the weight value into the regression equation:
MPG = 43.32625 - 0.600702 * 20
Calculating this, we get:
MPG = 43.32625 - 0.600702 * 20
MPG = 43.32625 - 12.01404
MPG = 31.31221
So, the estimated MPG for a car weighing 20 hundred pounds is approximately 31.31221.
To estimate the MPG for a car that weighs 60 hundred pounds, substitute the weight value into the regression equation:
MPG = 43.32625 - 0.600702 * 60
Calculating this, we get:
MPG = 43.32625 - 0.600702 * 60
MPG = 43.32625 - 36.04212
MPG = 7.28413
So, the estimated MPG for a car weighing 60 hundred pounds is approximately 7.28413.